Abstract
For periodic graph operators, we establish criteria to determine the overlaps of spectral band functions based on Bloch varieties. One criterion states that for a large family of periodic graph operators, the irreducibility of Bloch varieties implies no non-trivial periods for spectral band functions. This particularly shows that spectral band functions of discrete periodic Schrödinger operators on ℤd have no non-trivial periods, answering positively a question asked by Mckenzie and Sabri [Quantum ergodicity for periodic graphs, Comm. Math. Phys. 403 (2023), 1477–1509].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Bättig, A Toroidal Compactification of the Two Dimensional Bloch-Manifold, Ph.D. thesis, ETH Zurich, 1988.
D. Battig, A toroidal compactification of the Fermi surface for the discrete Schrödinger operator, Comment. Math. Helv. 67 (1992), 1–16.
D. Bättig, H. Knorrer and E. Trubowitz, A directional compactification of the complex Fermi surface, Compositio Math. 79 (1991), 205–229.
G. Berkolaiko, Y. Canzani, G. Cox and J. L. Marzuola, A local test for global extrema in the dispersion relation of a periodic graph, Pure Appl. Anal. 4 (2022), 257–286.
N. Do, P. Kuchment and F. Sottile, Generic properties of dispersion relations for discrete periodic operators, J. Math. Phys. 61 (2020), Article no. 103502.
M. Embree and J. Fillman, Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials, J. Spectr. Theory 9 (2019), 1063–1087.
M. Faust and J. Lopez-Garcia, Irreducibility of the dispersion relation for periodic graphs, arXiv:2302.11534 [math.AG]
M. Faust and F. Sottile, Critical points of discrete periodic operators, arXiv:2206.13649 [math-ph]
J. Fillman and R. Han, Discrete Bethe–Sommerfeld conjecture for triangular, square, and hexagonal lattices, J. Anal. Math. 142 (2020), 271–321.
J. Fillman, W. Liu and R. Matos, Irreducibility of the Bloch variety for finite-range Schrödinger operators, J. Funct. Anal. 283 (2022), Article no. 109670.
J. Fillman, W. Liu and R. Matos, Algebraic properties of the Fermi variety for periodic graph operators, J. Funct. Anal. 286 (2024), no. 4, Paper No. 110286.
N. Filonov and I. Kachkovskiy, On spectral bands of discrete periodic operators, Comm. Math. Phys. 405 (2024), no. 2, Paper no. 21.
L. Fisher, W. Li and S. P. Shipman, Reducible Fermi surface for multi-layer quantum graphs including stacked graphene, Comm. Math. Phys. 385 (2021), 1499–1534.
D. Gieseker, H. Knörrer and E. Trubowitz, The Geometry of Algebraic Fermi Curves, Academic Press, Boston, MA, 1993.
R. Han and S. Jitomirskaya, Discrete Bethe–Sommerfeld conjecture, Comm. Math. Phys. 361 (2018), 205–216.
H. Knörrer and E. Trubowitz, A directional compactification of the complex Bloch variety, Comment. Math. Helv. 65 (1990), 114–149.
C. Kravaris, On the density of eigenvalues on periodic graphs, SIAM J. Appl. Algebra Geom. 7 (2023), 585–609.
P. Kuchment, An overview of periodic elliptic operators, Bull. Amer. Math. Soc. (N.S.) 53 (2016), 343–414.
W. Li and S. P. Shipman, Irreducibility of the Fermi surface for planar periodic graph operators, Lett. Math. Phys. 110 (2020), 2543–2572.
W. Liu, Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues, Geom. Funct. Anal. 32 (2022), 1–30.
Wencai Liu, Topics on Fermi varieties of discrete periodic Schrödinger operators, J. Math. Phys. 63 (2022), Article no. 023503.
W. Liu, Fermi isospectrality of discrete periodic Schrödinger operators with separable potentials on ℤ2, Comm. Math.Phys. 399 (2023), 1139–1149.
W. Liu, Fermi isospectrality for discrete periodic Schrödinger operators, Comm. Pure Appl. Math. 77 (2024), 1126–1146.
W. Liu, Floquet isospectrality for periodic graph operators, J. Differential Equations 374 (2023), 642–653.
T. Mckenzie and M. Sabri, Quantum ergodicity for periodic graphs, Comm. Math. Phys. 403 (2023), 1477–1509.
M. Sabri and P. Youssef, Flat bands of periodic graphs, J. Math. Phys. 64 (2023), Article no. 092101.
S. P. Shipman, Reducible Fermi surfaces for non-symmetric bilayer quantum-graph operators, J. Spectr. Theory 10 (2020), 33–72.
Acknowledgments
This research was supported by NSF DMS-2000345, DMS-2052572 and DMS-2246031. I would like to express my sincere gratitude to Mostafa Sabri for drawing my attention to Question 1 (see footnote 1), many valuable discussions on this subject and comments on earlier versions of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, W. Bloch varieties and quantum ergodicity for periodic graph operators. JAMA (2024). https://doi.org/10.1007/s11854-024-0339-y
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s11854-024-0339-y